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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation}
y^{\prime \prime}+b y^{\prime}+c y=0.\tag{3.4.1}
\end{equation}
</div>
<p class="continuation">Seek a solution of the form:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
y=e^{r x}.
\end{equation*}
</div>
<p class="continuation">The characteristic equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
r^2+b r+c=0.
\end{equation*}
</div>
<p class="continuation">We consider the case it has two complex conjugate roots, say</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
r_1=\lambda+i \mu,\quad r_2=\lambda-i \mu, \quad \mu \neq 0.
\end{equation*}
</div>
<p class="continuation">There are two solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
\begin{aligned}
&amp;y_1=e^{(\lambda+i\mu) x}=e^{\lambda x} e^{i \mu x}=e^{\lambda x} (\cos \mu x+i \sin \mu x)\\
&amp;y_2=e^{(\lambda-i\mu) x}=e^{\lambda x} e^{-i \mu x}=e^{\lambda x} (\cos \mu x-i \sin \mu x).
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Any linear combination of <span class="process-math">\(y_1\)</span> and <span class="process-math">\(y_2\)</span> are still solutions, therefore,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
\begin{aligned}
&amp;y_{11}=\frac{1}{2} y_1+\frac{1}{2} y_2=e^{\lambda x} \cos \mu x,\\
&amp;y_{12}=\frac{1}{2 i}\left( y_1- y_2 \right)=e^{\lambda x} \sin \mu x,
\end{aligned}
\end{equation*}
</div>
<p class="continuation">are also two solutions to (<a href="" class="xref" data-knowl="./knowl/eq3_10.html" title="Equation 3.4.1">(3.4.1)</a>). Consider the Wronskian of <span class="process-math">\(y_{11}\)</span> and <span class="process-math">\(y_{12}\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
\begin{aligned}
W(y_{11}, y_{12})&amp;=\left|
\begin{array}{ll}
y_{11} &amp; y_{12}\\
y_{11}^{\prime} &amp; y_{12}^{\prime}
\end{array}
\right|\\
&amp;=\left|
\begin{array}{lll}
e^{\lambda x} \cos \mu x &amp; \quad &amp; e^{\lambda x} \sin \mu x\\
\lambda e^{\lambda x} \cos \mu x-\mu e^{\lambda x}\sin \mu x &amp; \quad &amp;
\lambda e^{\lambda x} \sin \mu x+\mu e^{\lambda x}\cos \mu x
\end{array}
\right|\\
&amp;=\mu e^{2 \lambda x} \neq 0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, <span class="process-math">\(y_{11}\)</span> and <span class="process-math">\(y_{12}\)</span> are a fundamental set of solutions and the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_10.html">
\begin{equation*}
y=C_1 e^{\lambda x} \cos \mu x+C_2 e^{\lambda x} \sin \mu x.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_4.html#p-97" class="internal">in-context</a></span>
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